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How to Use Bernoulli's Equation for Inviscid Flow

Key Takeaways

  • What exactly is Bernoulli’s equation?

  • Applying Bernoulli’s equation to different flow types.

  • Euler’s and Bernoulli’s equation inviscid fluid flow solution.

Bernoulli’s principle

Illustration of Bernoulli’s equation for fluid flow

With the speed at which technology changes these days, it is not surprising that many people–with due deference to historians and grandparents–spend little time contemplating the past. Yet, virtually all epiphanies or significant discoveries, especially in science, are due to the efforts of figures from the past. This is most aptly manifested in the scientific literature that serves as the cumulative and permanent record of important achievements such as the postulation and experimental verification of new theories. 

Fortunately, the greatest contributors to our scientific knowledge clearly recognized the importance of this chain of succession. A good example of this is Bernoulli’s principle, which is derivable from the Law of the Conservation of Energy–first proposed by Émilie du Châtelet–that is based upon Isaac Newton’s Second Law of Motion. However, the application of this important principle via Bernoulli’s equation for inviscid flow was made possible by Leonhard Euler.   

What Is Bernoulli’s Equation?

Prior to defining Bernoulli’s equation for fluid flow—or Bernoulli’s equations (see the section below)—it is important to understand Bernoulli’s principle, which can be defined as follows:

Bernoulli’s principle says that in a fluid flow streamline moving in the horizontal direction, points with higher flow velocities will have lower pressures than points with slower fluid velocities.  

We can see this principle in action by observing the airflow around a cambered airfoil as an aircraft attains lift during takeoff, shown in the figure below. 

Bernoulli’s principle example in aerodynamics

Example of Bernoulli’s principle

As shown in the figure above, Bernoulli’s principle accurately explains aerodynamic lift dynamics due to the relationships between fluid velocities and pressures. This principle stated mathematically is the Bernoulli equation for fluid flow, shown below.

General Bernoulli’s equation for fluid flow

Bernoulli’s equation for fluid flow

In the above equation:

    P denotes pressure

    𝛒 denotes density

    𝝼 denotes velocity

    g denotes gravity 

    h denotes height

This equation requires that fluid flow be isentropic and inviscid–or the viscosity is or can be assumed to be zero–and can be applied to various types of fluid flow. For example, for water through a pipe as well as airflow at a boundary layer in aerodynamics

Applying Bernoulli’s Equation to Different Flow Types

The equation defined in the previous section is commonly referred to as Bernoulli’s equation, which can be derived from the Navier-Stokes equation. This equation is applicable to incompressible fluid flow–low Mach number, which is the ratio of flow velocity to the speed of sound–and can be written in the following equivalent form.

Bernoulli’s Equation for Incompressible Flow:

Alternative form of Bernoulli’s equation for incompressible flow

Simplified form of Bernoulli’s equation for incompressible flow

The above equation is derived by simply multiplying both sides by the density, subtracting gh1 from both sides, and replacing the RHS with a constant. Note that the constant only applies to the fluid flow system under study. Bernoulli’s equation can also be applied to compressible fluid flows–most fluids are compressible. The equation for the latter is below.

Bernoulli’s Equation for Compressible Flow:

Form of Bernoulli’s equation for compressible fluid flow

 Bernoulli’s equation for compressible flow

In the equation above, 𝞊 is the thermodynamic internal energy. Irrespective of the form, Bernoulli’s equation is applicable only for inviscid flow and is derivable from Euler’s equation of motion.

Euler’s and Bernoulli’s Equation Inviscid Fluid Flow Solution

Bernoulli’s equation for inviscid flow is applicable to many fluid flow problems. However, there is another equation that is often used to analyze fluid systems where viscosity is negligible. That is Euler’s momentum equation for inviscid flow, shown below.

. Alternative to Bernoulli’s equation for inviscid flow

Euler’s equation for inviscid flow

The utilization of the above Euler’s equation requires that the following conditions can be assumed:

  • Fluid is incompressible.
  • Flow is non-turbulent or most likely laminar.
  • The velocity is constant.
  • There is no viscosity.
  • The dominant forces acting on the fluid are gravity and pressure. 

Similarly, there are requirements for the use of Bernoulli’s equation, defined above, for inviscid flow:

  • Fluid is incompressible.
  • Flow is irrotational or non-turbulent.
  • The velocity is the same over a cross-sectional area.
  • There is no viscosity.
  • The dominant forces acting on the fluid are gravity and pressure. 
  • Only applicable along a streamline–imaginary line tangential to the flow velocity vector.

From these lists of assumptions, it is obvious that both equations can be applied to similar problems. The best way to apply the Euler or Bernoulli equation inviscid flow solution is by employing an advanced CFD tool such as Cadence’s Omnis, which allows for the modeling and comparison of different flow scenarios. And, if your system design includes aerodynamic analysis, a powerful grid generation tool such as Pointwise, which is integrated into Cadence’s CFD tools, is also necessary. 

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